Axonemes form the core of eukaryotic flagella and cilia, performing tasks ranging from transporting fluid in developing embryos to the propulsion of sperm. Despite their abundance across the eukaryotic domain, the mechanisms that regulate the beating action of axonemes remain unknown. The flagellar waveforms are 3D in general, but current understanding of how axoneme components interact stems from 2D data; comprehensive measurements of flagellar shape are beyond conventional microscopy. Moreover, current flagellar model systems (e.g., sea urchin, human sperm) contain accessory structures that impose mechanical constraints on movement, obscuring the "native" axoneme behavior. We address both problems by developing a high-speed holographic imaging scheme and applying it to the (male) microgametes of malaria (Plasmodium) parasites. These isolated flagella are a unique, mathematically tractable model system for the physics of microswimmers. We reveal the 3D flagellar waveforms of these microorganisms and map the differential shear between microtubules in their axonemes. Furthermore, we overturn claims that chirality in the structure of the axoneme governs the beat pattern [Hirokawa N, et al. (2009) Ann Rev Fluid Mech 41:53-72], because microgametes display a left- or right-handed character on alternate beats. This breaks the link between structural chirality in the axoneme and larger scale symmetry breaking (e.g., in developing embryos), leading us to conclude that accessory structures play a critical role in shaping the flagellar beat.